Integral of the Clarke subdifferential mapping and a generalized Newton–Leibniz formula

Abstract

In the theory of Lebesgue integration it has been proved that if f is a real Lipschitz function defined on a segment [ a , b ] ⊂ R , then the Newton–Leibniz formula ∫ a b f ′ ( x ) d x = f ( b ) − f ( a ) (the fundamental theorem of calculus) holds. This paper extends the fact to the case where the Fréchet derivative f ′ ( ⋅ ) (which is defined almost everywhere on [ a , b ] by the Rademacher theorem) and the Lebesgue integral are replaced, respectively, by the Clarke subdifferential mapping ∂ C f ( ⋅ ) and the Aumann (set-valued) integral. Among other things, we show that f ( b ) − f ( a ) ∈ ∫ a b ∂ C f ( x ) d x and the equality ∫ a b ∂ C f ( x ) d x = { f ( b ) − f ( a ) } is valid if and only if f is strictly Hadamard differentiable almost everywhere on [ a , b ] . The result is derived from a general representation formula, which we obtain herein for the integral of the Clarke subdifferential mapping of a Lipschitz function defined on a separable Banach space.